Procedural square shape in Cycles (with similar effect to Gaussian blur)
Here's an alternative solution using maths nodes to calculate distance from the square and the Gaussian distribution function.



The key here is for any point in the shader to calculate the closest point to the square. The square is defined by 4 boundary values - X-Min, X-Max, Y-Min, Y-Max.

To calculate the closest point that is still within the square we can simply use the Max and Min functions. ie,

closest-X = Min( Max(X, X-Min), X-Max)
closest-Y = Min( Max(Y, Y-Min), Y-Max)


To calculate the distance to this point (ie, the closest distance to the square) we can use Pythagoras

distance^2 = X^2 + Y^2


ie,

distance = sqrt(X^2 + Y^2)


For the Gaussian falloff we use the equation :

 intensity = 1 / (2^dist)


Bringing it all together produces the following material (note that I've omitted the 'sqrt' of the distance calculation - this gives a more pronounced falloff based on the square of the distance, rather than the plain distance) :



Adjusting the input nodes (X-Min, X-Max, Y-Min, Y-Max, Focus) will adjust the size and 'blurriness' of the square. Note that the Gaussian blur starts from the edge of the square - so the square is always maximum intensity with the falloff starting at its boundary. This means that the blur will always make it appear larger - adjust the boundary values to compensate whilst adjusting the Focus if this is not desired.

To tile multiple squares you can simply use a Mapping node and Modulo nodes to repeat the range multiple times :



Animating the mapping and various inputs can produce the following result :



Blend file attached 



This can be used to generate procedural tiles as shown by this sample image provided by @cgslav :